OFFSET
1,33
COMMENTS
Number of partitions of n into such odd parts that the sum of their reciprocals is one. - Antti Karttunen, Jul 23 2018
It would be nice to know whether nonzero values may occur only on n of the form 8k+1.
LINKS
David A. Corneth, Table of n, a(n) for n = 1..370 (terms 1..150 from Seiichi Manyama, terms 150..273 from Antti Karttunen)
David A. Corneth, Tuples up to n = 370.
FORMULA
a(2*k) = 0. - David A. Corneth, Jul 24 2018
EXAMPLE
1 = 1/3 + 1/3 + 1/3, the sum of denominators is 9, this is the only expression of 1 as unit fractions with odd denominators that sum to 9, so a(9)=1.
1 = 1/15 + 1/5 + 1/5 + 1/5 + 1/3 = 1/9 + 1/9 + 1/9 + 1/3 + 1/3 are the only solutions with odd denominators that sum to 33, thus a(33) = 2. - Antti Karttunen, Jul 24 2018
MATHEMATICA
Array[Count[IntegerPartitions[#, All, Range[1, #, 2]], _?(Total[1/#] == 1 &)] &, 70] (* Michael De Vlieger, Jul 26 2018 *)
PROG
(Ruby)
def f(n)
n - 1 + n % 2
end
def partition(n, min, max)
return [[]] if n == 0
[f(max), f(n)].min.step(min, -2).flat_map{|i| partition(n - i, min, i).map{|rest| [i, *rest]}}
end
def A270599(n)
ary = [1]
(2..n).each{|m|
cnt = 0
partition(m, 2, m).each{|ary|
cnt += 1 if ary.inject(0){|s, i| s + 1 / i.to_r} == 1
}
ary << cnt
}
ary
end
(PARI) A270599(n, maxfrom=n, fracsum=0) = if(!n, (1==fracsum), my(s=0, tfs, k=(maxfrom-!(maxfrom%2))); while(k >= 1, tfs = fracsum + (1/k); if(tfs > 1, return(s), s += A270599(n-k, min(k, n-k), tfs)); k -= 2); (s)); \\ Antti Karttunen, Jul 23 2018
(PARI)
\\ More verbose version for computing values of a(n) for large n:
A270599(n) = if(!(n%2), 0, my(s=0); forstep(k = n, 1, -2, print("A270599(", n, ") at toplevel, k=", k, " s=", s); s += A270599aux(n-k, min(k, n-k), 1/k)); (s));
A270599aux(n, maxfrom, fracsum) = if(!n, (1==fracsum), my(s=0, tfs, k=(maxfrom-!(maxfrom%2))); while(k >= 1, tfs = fracsum + (1/k); if(tfs > 1, return(s), s += A270599aux(n-k, min(k, n-k), tfs)); k -= 2); (s)); \\ Antti Karttunen, Jul 24 2018
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Mar 26 2016
EXTENSIONS
Name corrected by Antti Karttunen, Jul 23 2018 at the suggestion of David A. Corneth
STATUS
approved